A characterization of regular tetrahedra in Z3

Size: px
Start display at page:

Download "A characterization of regular tetrahedra in Z3"

Transcription

1 Columbus State University CSU epress Faculty Bibliography 2009 A characterization of regular tetrahedra in Z3 Eugen J. Ionascu Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Ionascu, Eugen J., "A characterization of regular tetrahedra in Z3" (2009). Faculty Bibliography. Paper This Article is brought to you for free and open access by CSU epress. It has been accepted for inclusion in Faculty Bibliography by an authorized administrator of CSU epress.

2 A characterization of regular tetrahedra in Z 3 Eugen J. Ionascu Department of Mathematics Columbus State University Columbus, GA 31907, US Honorific Member of the Romanian Institute of Mathematics Simion Stoilow ionascu eugen@colstate.edu December 23 rd, 2007 Abstract In this note we characterize all regular tetrahedra whose vertices in R 3 have integer coordinates. The main result is a consequence of the characterization of all equilateral triangles having integer coordinates ([3]). Previous work on this topic began in [4]. We will use this characterization to point out some corollaries. The number of such tetrahedra whose vertices are in the finite set {0, 1, 2,..., n} 3, n N, is related to the sequence A in the Online Encyclopedia of Integer Sequences ([9]). 1 Introduction In this paper we give a solution to the following system of Diophantine equations: (x a x b ) 2 + (y a y b ) 2 + (z a z b ) 2 = (x a x c ) 2 + (y a y c ) 2 + (z a z c ) 2 (x a x b ) 2 + (y a y b ) 2 + (z a z b ) 2 = (x a x d ) 2 + (y a y d ) 2 + (z a z d ) 2 (x a x b ) 2 + (y a y b ) 2 + (z a z b ) 2 = (x b x c ) 2 + (y b y c ) 2 + (z b z c ) 2 (1) (x a x b ) 2 + (y a y b ) 2 + (z a z b ) 2 = (x b x d ) 2 + (y b y d ) 2 + (z b z d ) 2 (x a x b ) 2 + (y a y b ) 2 + (z a z b ) 2 = (x c x d ) 2 + (y c y d ) 2 + (z c z d ) 2 where x a, y a, z a, x b, y b, z b, x c, y c, z c, x d, y d, z d are in Z. If A(x a, y a, z a ), B(x b, y b, z b ), C(x c, y c, z c ), and D(x d, y d, z d ) are considered in R 3 the Diophantine system (1) represents exactly the condition that makes ABCD a regular tetrahedron with vertices having integer coordinates. Clearly (1) is invariant under translations in R 3 with vectors having integer 1

3 coordinates. So, without loss in generality we are interested in the case, A = O, where O is the origin. Then for each of the faces OBC, OCD and OBD to be being equilateral triangles it is necessary and sufficient to solve for instance the following system: x 2 b + y2 b + z2 b = x2 c + yc 2 + zc 2 (2) x 2 b + y2 b + z2 b = (x b x c ) 2 + (y b y c ) 2 + (z b z c ) 2. This problem was solved in [3]. We recall the following facts from [3] and [4]. Every equilateral triangle in R 3 whose vertices have integer coordinates, after a translation that brings one of its vertices to the origin, must be contained in a plane of a normal vector (a, b, c) (a, b, c Z) satisfying the Diophantine equation a 2 + b 2 + c 2 = 3d 2 (3) with d N. Moreover, a, b, c, and d can be chosen so that the sides of such a triangle have length of the form d 2(m 2 mn + n 2 ) with m and n integers. A more precise statement is the next parametrization of these triangles. Theorem 1.1. Let a, b, c, d be odd integers such that a 2 + b 2 + c 2 = 3d 2 and gcd(a, b, c) = 1. Then for every m, n Z the points P (u, v, w) and Q(x, y, z) whose coordinates are given by u = m u m n u n, x = m x m n x n, v = m v m n v n, and y = m y m n y n, (4) w = m w m n w n, z = m z m n z n, with m x = 1 [db(3r + s) + ac(r s)]/q, n 2 x = (rac + dbs)/q m y = 1 [da(3r + s) bc(r s)]/q, 2 n y = (das bcr)/q m z = (r s)/2, n z = r and m u = (rac + dbs)/q, n u = 1 [db(s 3r) + ac(r + s)]/q 2 m v = (das rbc)/q, n v = 1 [da(s 3r) bc(r + s)]/q 2 m w = r, n w = (r + s)/2 where q = a 2 + b 2 and (r, s) is a suitable solution of 2q = s 2 + 3r 2 that makes all the numbers in (5) integers, together with the origin (O(0,0,0)) forms an equilateral triangle in Z 3 contained in the plane having equation P a,b,c := {(α, β, γ) R aα + bβ + cγ = 0} and having sides-lengths equal to d 2(m 2 mn + n 2 ). Conversely, there exists a choice of the integers r and s such that given an arbitrary equilateral triangle in R 3 whose vertices, one at the origin and the other two having integer coordinates and contained in the plane P a,b,c, then there also exist integers m and n such that the two vertices not at the origin are given by (4) and (5). 2 (5)

4 Figure 1: Lattice generated by (4) with m, n Z in the plane P a,b,c := {(α, β, γ) R aα + bβ + cγ = 0} In Figure 1 we are showing a few triangles that can be obtained, using (4), for various values of m and n. Out of all these equilateral triangles only a few are faces of regular tetrahedra with integer coordinates. The restriction comes from the following characterization given as Proposition 5.3 in [4]. Proposition 1.1. A regular tetrahedron having side lengths l and with integer coordinates exists, if and only if l = m 2 for some m N. Given an m N, we are interested in finding a characterization of all such tetrahedra having side lengths equal to m 2. Let us denote this class of objects by T m and its subset of tetrahedra that have the origin as a vertex by T 0 m. Obviously we have T m = v Z 3(T0 m + v) but this union is not a partition. In what follows we will use the notation T := (T m ). m N This latter union is, on the other hand, a partition. Next we are going to concentrate on T 0 m. It turns out that T 0 m is numerous if m is divisible by many prime factors of the form 6k + 1, k N, as one can see from the following. First we recall Euler s 6k + 1 theorem (see [7], pp. 568 and [1], pp. 56). Proposition 1.2. An integer t can be written as s 2 sr + r 2 for some s, r Z if and only if in the prime factorization of t, 2 and the primes of the form 6k 1 appear to an even exponent. The following result seems to be known but we could not find a reference for it. Proposition 1.3. Given k = 3 α (a 2 )b where a does not contain any prime factors of the form 6l + 1, b is the product of such primes, say b = p r 1 1 p r 2 2 p r j j, then the number of representations k = m 2 mn + n 2, counting all possible pairs (m, n) of integers that satisfy this equation, is equal to 6(r 1 + 1)(r 2 + 1) (r j + 1). 3

5 For a proof of this proposition one can use arguments similar to those in the proof of Gauss s Theorem about the number of representations of a number as sums of two squares of integers ([7]). In this case one has to replace Gaussian integers with Eisenstein integers, i.e. the ring of numbers of the form m + nω, m, n Z, where ω = 1+i 3. Related to 2 Proposition 1.3, we found a conjecture in a paper posted on the mathematics archives (see [6]). This article refers to the number of representations of a number as m 2 + mn + n 2 with m, n N. 2 Main results Let us begin by refining the argument used in the proof of Proposition 5.3 in [4]. That proposition referred to the following figure representing a regular tetrahedron whose vertices are given by the origin, P (x, y, z), Q(u, v, w) and R: Figure 2: Regular tetrahedron By the characterization of equilateral triangles in Theorem 1.1 we may assume that the coordinates of P and Q are given by (4) and (5). If E denotes the center of the face OP Q. Then from Theorem 1.1 we know that ER ER = (a,b,c) a 2 +b 2 +c 2 = 1 3d (a, b, c) for some a, b, c, d, m, n Z, gcd(a, b, c) = 1, d odd satisfying a 2 + b 2 + c 2 = 3d 2 and l = d 2(m 2 mn + n 2 ). Let us denote m 2 mn + n 2 by ζ(m, n). The coordinates of E are ( u+x, y+v, z+w) From the Pythagorean theorem one can find easily that RE = l. Since 3 OR = OE + ER, the coordinates of R must be given by ( u + x 2 1 ± l a, y + v 2 1 ± l b, z + w ) 2 1 ± l c 3 3 3d 3 3 3d 3 3 3d 4

6 or ( u + x 3 ± 2a ζ(m, n), y + v 3 3 ) ± 2b ζ(m, n), z + w ± 2c ζ(m, n) Since these coordinates are assumed to be integers we see that ζ(m, n) must be a perfect square, say k 2, k N. It is worth mentioning that this leads to the Diophantine equation m 2 mn + n 2 = k 2 (6) whose positive solutions are known as Eisenstein triples, or Eisenstein numbers. A primitive solution of (6) is one for which gcd(m, n) = 1. These triples can be characterized in a similar way as the Pythagorean triples. Theorem 2.1. Every primitive solution of the Diophantine equation (6) is in one of the two forms: m = v 2 u 2, n = 2uv u 2 and k = u 2 uv + v 2, with v > u, or m = 2uv u 2, n = 2uv v 2 and k = u 2 uv + v 2, with 2v > u > v/2, where u, v N, gcd(u, v) = 1, and u + v 0 (mod 3). Conversely, every triple given by one of the alternatives in (7) is a primitive solution of (6). We leave the proof of this theorem to the reader. Next, let us introduce the notation Ω(k) := {(m, n) Z 2 : ζ(m, n) = k 2 }. If the primes dividing k are all of the form p 2 (mod 3) then we have simply Ω(k) = {(k, 0), (k, k), (0, k), ( k, 0), ( k, k), (0, k)} but in general this set can be a much larger as one can see from Proposition 1.3. For instance if k = 7 then one can check the nontrivial representation 49 = ζ(8, 3). For each solution (m, n) Ω(k), in general, one can find eleven more by applying the following transformations: (m, n) τ 1 (m n, m), (m, n) τ 2 (m, n m), then their permutations (m, n) τ 3 (n, m), (m, n) τ 4 (m, m n), (m, n) τ 5 (n m, n), and finally the reflections into the origin of all the above maps (m, n) τ 6 ( m, n), (m, n) τ 7 (n m, m), (m, n) τ 8 ( m, m n), (m, n) τ 9 ( n, m), (m, n) τ 10 ( m, n m), (m, n) τ 11 (m n, n). Hence, the coordinates of R depend on (m, n) Ω(k) and two possible choices of signs: R = (m x + m u )m (n x + n u )n ±2ak 3, (m y + m v )m (n y + n v )n ±2bk 3, (m z + m w )m (n z + n w )n ±2ck 3 (7), (m, n) Ω(k). (8) We would like to show that for every primitive solution of the equation (3), every k N, and every (m, n) Ω(k) one has either integer coordinates in (8) or can choose the signs in 5

7 order to accomplish this. A primitive solution of (3) is a solution that satisfies the conditions gcd(a, b, c) = 1. We begin with a few lemmas. Lemma 2.1. Every primitive solution (a, b, c, d) of (3) must satisfy (i) a ±1 (mod 6), b ±1 (mod 6) and c ±1 (mod 6), (ii) a 2 + b 2 2 (mod 6). Proof. One can see that all numbers a, b, c and d must be odd since gcd(a, b, c) = 1. Then an odd perfect square is congruent to only 1 or 3 modulo 6. By way of contradiction if we have for instance a 3 (mod 6) then 3 divides a 2 = 3d 2 (b 2 + c 2 ). This implies that 3 divides b 2 + c 2 which is possible only if 3 divides b and c. Therefore that would contradict the assumption gcd(a, b, c) = 1. Hence, (i) is shown and (ii) is just a simple consequence of (i). Let us observe that (ii) in Lemma 2.1 implies in particular that q = a 2 + b 2 is coprime with 3. Lemma 2.2. Let k N and (m, n) Ω(k). (a) Then m and n must satisfy k ±(m + n) (mod 3). (b) If k 0 (mod 3) then m n 0 (mod 3). Proof. (a) This part follows immediately from the identities k 2 4k 2 = 4m 2 4mn + 4n 2 = 3m 2 + (2m n) 2 (2m n) 2 (m + n) 2 (mod3). (b) For the second part let us observe that if k 0 (mod 3) then by part (a) we must have m = 3t n for some t Z. This in turn gives 9k 2 = k 2 = m 2 mn + n 2 = 3(3t 3 3nt + n 2 ) which shows that 3 divides 3t 3 3nt + n 2. So, finally n must be divisible by 3 and then so is m. Figure 3: All regular tetrahedra in T 0 1 Lemma 2.3. Let q = a 2 + b 2 with (a, b, c, d) a primitive solution of (3) and r, s Z a solution of the equation 2q = s 2 + 3r 2. Then 1 a 2 b 2 c 2 s 2 (mod 3). 6

8 Proof. Since 2c 2 = 6d 2 2q s 2 3r 2 s 2 (mod 3) then we have 2c 2 2s 2 (mod 3) which together with Lemma 2.1 gives the desired conclusion. For a = b = c = 1, we have 8 elements in T 0 1 as shown in Figure 3, one for each triangular face of a cuboctahedron (Archimedean solid A 1 or also known as a truncated cube): The number of tetrahedra in T 0 m in general depends on m. For instance, if m = 3 then we calculated that T 0 3 = 40. Let us denote by T O(m) the cardinality of T 0 m. In general, the faces of an element in T 0 m must have normal vectors that satisfy (3). But these equations can be very different and one has to go search far enough to find such an example. For instance, the tetrahedron OABC where A = (376, 841, 2265), B = ( 1005, 2116, 701), C = (1411, 1965, 356) has the four faces with normal vectors ( 187, 113, 73), satisfying = 3(133 2 ), ( 343, 253, 37), satisfying = 3(247) 2, (19, 41, 151), satisfying = 3(91) 2 and (391, 2461, 1661), satisfying = 3(1729) 2. Theorem 2.2. Every tetrahedra in T 0 l can be obtained by taking as one of its faces an equilateral triangle having the origin as a vertex and the other two vertices given by (4) and (5) with a, b, c and d odd integers satisfying (3) with d a divisor of l, and then completing it with the fourth vertex as in (8) for some (m, n) Ω(l/d). Conversely, if we let a, b, c and d be a primitive solution of (3), let k N and (m, n) Ω(k), then the coordinates of the point R in (8) are (a) all integers, if k 0 (mod 3) regardless of the choice of signs or (b) integers, precisely for only one choice of the signs if k 0 (mod 3). Proof. The first part of the theorem follows from the arguments at the beginning of this section. For the second part, the case in which k is a multiple of 3 follows from Lemma 2.2. Let us look into a stituation in which k 0 (mod 3). First we are going to analyze the third coordinate of R. Since (5) gives m z + m w = r s + r = 3r s and n 2 2 z + n w = 3r+s we see 2 that 2[(m z + m w )m (n z + n w )n ± 2ck] s(m + n) ± ck (mod 3), (9) which by Lemma 2.2 and Lemma 2.1 give that s(m + n) ± ck ±k(s ± c) 0 (mod 3) only for one choice of signs. So, the last coordinate of R satisfies the statement of our theorem in part (b). If the coordinates of R are (x R, y R, z R ) we must have x 2 R + y2 R + z2 R = l 2. So, it is enough to analyze just one other coordinate of R. From (5) we obtain that 2q(m x + m u ) = db(3r + s) ac(r s) 2(rac + dbs) acs (mod 3), and 2q(n x + n u ) = 2(rac + dbs) db(s 3r) ac(r + s) acs (mod 3). Therefore, 2q[(m x + m u )m (n x + n u )n ± 2ak] acs(m + n) ± qak (mod 3). 7

9 But q 1 (mod 3) by Lemma 2.1, and so 2q[(m x + m u )m (n x + n u )n ± 2ak] a(cs(m + n) k) (mod 3). Because c 2 1 (mod 3), we can multiply the above relation by c to get 2qc[(m x + m u )m (n x + n u )n ± 2ak] a[ s(m + n) ± ck] (mod 3), which shows, based on (9), that (m x + m u )m (n x + n u )n ± 2ak is divisible by 3 if and only if z R is an integer. Hence we have proved the last part of Theorem 2.2. The next figure shows how our tetrahedra look if we start from equilateral triangles as bases contained in a plane and this is as expected according to Theorem 2.2. Figure 4: Regular tetrahedra in the case k 0 (mod 3) Remark: The alternating behavior that can be observed in Figure 4 follows from Theorem 2.2 part (b). Indeed, we can start with an equilateral triangle contained in a plane P which is one of the tiles and the generator of a regular triangular tessellation. By Theorem 2.2 part (b) each such tile from this tessellation is the face of a regular tetrahedron in T that is located in one and only one of the sides of P. We denote the two half spaces by P + and P. Let us take one such tetrahedron, say in P +, and translate this tetrahedron such that the fourth vertex, R, becomes the origin. The other vertices give rise to three vertices of integer coordinates in P that are each the fourth vertex R for three other tetrahedra living in P as in Figure 4. This type of translation can be repeated with one of the tetrahedra in P and one obtains the alternating tetrahedra as Figure 4. Proposition 2.1. For a regular tetrahedra in T m each face lies in a plane with a normal vector n i = (a i, b i, c i ) Z 3, i = 1, 2, 3, 4, which satisfy a 2 i + b 2 i + c 2 i = 3d 2 i, d i an odd integer dividing m, 1 i 4 a i a j + b i b j + c i c j + d i d j = 0, 1 i < j 4 (10) 8

10 Proof. We are going to refer to the Figure 2. Let E be the center of the face OP Q, F be the center of the face RP Q, G the midpoint of the side P Q, and H the center of the tetrahedron ORP Q (O being the origin). Since O, R, E, F, G, and H are all coplanar, and F and E are the intersections of the medians on each corresponding face we have RF = 2 RG 3 and EG = 1 HO. By Menelaus s theorem RF EG HO = 1. This gives = 3 and from here EO 2 HF RG EO HF we have HO = HR = 3 2 l = 3l. Using the law of cosines in the triangle ORH gives: OR 2 = 2HR 2 2HR 2 cos ÔHR which gives cos ÔHR = 1 3. (11) So, if the normal exterior unit vectors of the faces RPQ and OPQ are, say (a 1,b 1,c 1 ) 3d1 and (a 2,b 2,c 2 ) 3d2, using (11) gives a 1 a 2 + b 1 b 2 + c 1 c 2 + d 1 d 2 = 0. The rest of the identities follow similarly or from what we have mentioned so far. Remark: If we consider the matrix MT := 1 2 a 1 b 1 c 1 d 1 d 1 d 1 1 a 2 b 2 c 2 d 2 d 2 d 2 1 a 3 b 3 c 3 d 3 d 3 d 3 1 a 4 b 4 c 4 d 4 d 4 d 4 1 (12) the restrictions in (10) can be reformulated in terms of MT by requiring it to be orthogonal, i.e. MT (MT ) t = (MT ) t MT = I. It is known that the set of orthogonal matrices form a group. We also want to point out that in (10) only two of the exterior normal vectors n i are essential. The other two can be computed from the given ones. An interesting question here is whether or not every two vectors (a, b, c) and (a, b, c ) which satisfy the conditions a 2 + b 2 + c 2 = 3d 2, a 2 + b 2 + c 2 = 3d 2, aa + bb + cc + dd = 0 (13) for some d, d Z, corresponds to a tetrahedron in T containing two faces normal to the given vectors. It turns out that the answer is a positive one and the proof of it follows from Theorem 4 in [4]. Another corollary of Theorem 2.2 about solutions of a particular case of the Diophantine system (13) is given next. Corollary 2.1. For every odd integer d > 1, the Diophantine system a 2 + b 2 + c 2 = 3d 2, a 2 + b 2 + c 2 = 3d 2, aa + bb + cc = d 2 (14) always has nontrivial solutions (one trivial solution is a = b = c = d and a = b = d, c = d.) Proof. We have shown in [4] that the Diophantine equation a 2 + b 2 + c 2 = 3d 2 always has nontrivial solutions. We then take m = n = 1 in Theorem 1.1 to obtain an equilateral 9

11 triangle contained in the plane of normal vector (a, b, c). By Theorem 2.2, case k = 1, we can complete this equilateral triangle to a regular tetrahedron in T. Taking the normals to the new faces will give four normals and say (â, ˆb, ĉ) is one of them. Let us assume that gcd(â, ˆb, ĉ) = 1. We know that there is a ˆd such that â 2 + ˆb 2 + ĉ 2 = 3 ˆd 2. The side length of this tetrahedron is, by our choice of m and n, equal to d 2. Therefore, from Theorem 1.1 applied to the plane normal to (â, ˆb, ĉ), we get that d 2 = ˆd 2(u 2 uv + v 2 ) for some u, v Z, which implies that ˆd divides d. Hence we can adjust (â, ˆb, ĉ), if necessary, by a multiplicative factor in order to satisfy (14). References [1] R. Guy, Unsolved Problems in Number Theory, Springer-Verlag, [2] E. Grosswald, Representations of Integers as Sums of Squares, Springer Verlag, [3] R. Chandler and E. J. Ionascu, A characterization of all equilateral triangles in Z 3, arxiv: v1, submitted December [4] E. J. Ionascu, A parametrization of equilateral triangles having integer coordinates, Journal of Integer Sequences, 10, (2007), Article [5] E. J. Ionascu, Counting all equilateral triangles in {0, 1,..., n} 3, to appear in Acta Mathematica Universitatis Comenianae [6] U. P. Nair, Elementary results on the binary quadratic form a 2 +ab+b 2, arxiv: v1 [7] K. Rosen, Elementary Number Theory, Fifth Edition, Addison Wesley, 2004 [8] C. J. Moreno and S.S. Wagstaff, JR. Sums of squares, Chapman & Hall/CRC, [9] Neil J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, 2005, published electronically at njas/sequences/ Mathematics Subject Classification: Primary 11A67; Secondary 11D09, 11D04, 11R99, 11B99, 51N20. Keywords: Diophantine equations, regular tetrahedra, equilateral triangles, integers, parametrization, characterization. (Concerned with sequence A ) 10

REGULAR TETRAHEDRA WHOSE VERTICES HAVE INTEGER COORDINATES. 1. Introduction

REGULAR TETRAHEDRA WHOSE VERTICES HAVE INTEGER COORDINATES. 1. Introduction Acta Math. Univ. Comenianae Vol. LXXX, 2 (2011), pp. 161 170 161 REGULAR TETRAHEDRA WHOSE VERTICES HAVE INTEGER COORDINATES E. J. IONASCU Abstract. In this paper we introduce theoretical arguments for

More information

A PARAMETRIZATION OF EQUILATERAL TRIANGLES HAVING INTEGER COORDINATES 1. INTRODUCTION

A PARAMETRIZATION OF EQUILATERAL TRIANGLES HAVING INTEGER COORDINATES 1. INTRODUCTION A PARAMETRIZATION OF EQUILATERAL TRIANGLES HAVING INTEGER COORDINATES EUGEN J. IONASCU arxiv:math/0608068v1 [math.nt] 2 Aug 2006 Abstract. We study the existence of equilateral triangles of given side

More information

A Parametrization of Equilateral Triangles Having Integer Coordinates

A Parametrization of Equilateral Triangles Having Integer Coordinates 1 3 47 6 3 11 Journal of Integer Sequences, Vol. 10 (007), Article 07.6.7 A Parametrization of Equilateral Triangles Having Integer Coordinates Eugen J. Ionascu Department of Mathematics Columbus State

More information

x y z 2x y 2y z 2z x n

x y z 2x y 2y z 2z x n Integer Solutions, Rational solutions of the equations 4 4 4 x y z x y y z z x n 4 4 and x y z xy xz y z n; And Crux Mathematicorum Contest Corner problem CC4 Konstantine Zelator P.O. Box 480 Pittsburgh,

More information

LATTICE PLATONIC SOLIDS AND THEIR EHRHART POLYNOMIAL

LATTICE PLATONIC SOLIDS AND THEIR EHRHART POLYNOMIAL Acta Math. Univ. Comenianae Vol. LXXXII, 1 (2013), pp. 147 158 147 LATTICE PLATONIC SOLIDS AND THEIR EHRHART POLYNOMIAL E. J. IONASCU Abstract. First, we calculate the Ehrhart polynomial associated with

More information

Ehrhart polynomial for lattice squares, cubes, and hypercubes

Ehrhart polynomial for lattice squares, cubes, and hypercubes Ehrhart polynomial for lattice squares, cubes, and hypercubes Eugen J. Ionascu UWG, REU, July 10th, 2015 math@ejionascu.ro, www.ejionascu.ro 1 Abstract We are investigating the problem of constructing

More information

SOME VARIANTS OF LAGRANGE S FOUR SQUARES THEOREM

SOME VARIANTS OF LAGRANGE S FOUR SQUARES THEOREM Acta Arith. 183(018), no. 4, 339 36. SOME VARIANTS OF LAGRANGE S FOUR SQUARES THEOREM YU-CHEN SUN AND ZHI-WEI SUN Abstract. Lagrange s four squares theorem is a classical theorem in number theory. Recently,

More information

Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing , People s Republic of China

Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing , People s Republic of China J. Number Theory 16(016), 190 11. A RESULT SIMILAR TO LAGRANGE S THEOREM Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing 10093, People s Republic of China zwsun@nju.edu.cn http://math.nju.edu.cn/

More information

M381 Number Theory 2004 Page 1

M381 Number Theory 2004 Page 1 M81 Number Theory 2004 Page 1 [[ Comments are written like this. Please send me (dave@wildd.freeserve.co.uk) details of any errors you find or suggestions for improvements. ]] Question 1 20 = 2 * 10 +

More information

2010 Fermat Contest (Grade 11)

2010 Fermat Contest (Grade 11) Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario 010 Fermat Contest (Grade 11) Thursday, February 5, 010

More information

NONABELIAN GROUPS WITH PERFECT ORDER SUBSETS

NONABELIAN GROUPS WITH PERFECT ORDER SUBSETS NONABELIAN GROUPS WITH PERFECT ORDER SUBSETS CARRIE E. FINCH AND LENNY JONES Abstract. Let G be a finite group and let x G. Define the order subset of G determined by x to be the set of all elements in

More information

Gaussian integers. 1 = a 2 + b 2 = c 2 + d 2.

Gaussian integers. 1 = a 2 + b 2 = c 2 + d 2. Gaussian integers 1 Units in Z[i] An element x = a + bi Z[i], a, b Z is a unit if there exists y = c + di Z[i] such that xy = 1. This implies 1 = x 2 y 2 = (a 2 + b 2 )(c 2 + d 2 ) But a 2, b 2, c 2, d

More information

On the prime divisors of elements of a D( 1) quadruple

On the prime divisors of elements of a D( 1) quadruple arxiv:1309.4347v1 [math.nt] 17 Sep 2013 On the prime divisors of elements of a D( 1) quadruple Anitha Srinivasan Abstract In [4] it was shown that if {1,b,c,d} is a D( 1) quadruple with b < c < d and b

More information

Lecture notes: Algorithms for integers, polynomials (Thorsten Theobald)

Lecture notes: Algorithms for integers, polynomials (Thorsten Theobald) Lecture notes: Algorithms for integers, polynomials (Thorsten Theobald) 1 Euclid s Algorithm Euclid s Algorithm for computing the greatest common divisor belongs to the oldest known computing procedures

More information

Determining elements of minimal index in an infinite family of totally real bicyclic biquadratic number fields

Determining elements of minimal index in an infinite family of totally real bicyclic biquadratic number fields Determining elements of minimal index in an infinite family of totally real bicyclic biquadratic number fields István Gaál, University of Debrecen, Mathematical Institute H 4010 Debrecen Pf.12., Hungary

More information

Proof 1: Using only ch. 6 results. Since gcd(a, b) = 1, we have

Proof 1: Using only ch. 6 results. Since gcd(a, b) = 1, we have Exercise 13. Consider positive integers a, b, and c. (a) Suppose gcd(a, b) = 1. (i) Show that if a divides the product bc, then a must divide c. I give two proofs here, to illustrate the different methods.

More information

OEIS A I. SCOPE

OEIS A I. SCOPE OEIS A161460 Richard J. Mathar Leiden Observatory, P.O. Box 9513, 2300 RA Leiden, The Netherlands (Dated: August 7, 2009) An overview to the entries of the sequence A161460 of the Online Encyclopedia of

More information

Fermat s Last Theorem

Fermat s Last Theorem Fermat s Last Theorem T. Muthukumar tmk@iitk.ac.in 0 Jun 014 An ancient result states that a triangle with vertices A, B and C with lengths AB = a, BC = b and AC = c is right angled at B iff a + b = c.

More information

Predictive criteria for the representation of primes by binary quadratic forms

Predictive criteria for the representation of primes by binary quadratic forms ACTA ARITHMETICA LXX3 (1995) Predictive criteria for the representation of primes by binary quadratic forms by Joseph B Muskat (Ramat-Gan), Blair K Spearman (Kelowna, BC) and Kenneth S Williams (Ottawa,

More information

HASSE-MINKOWSKI THEOREM

HASSE-MINKOWSKI THEOREM HASSE-MINKOWSKI THEOREM KIM, SUNGJIN 1. Introduction In rough terms, a local-global principle is a statement that asserts that a certain property is true globally if and only if it is true everywhere locally.

More information

0 Sets and Induction. Sets

0 Sets and Induction. Sets 0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set

More information

MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences.

MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. Congruences Let n be a postive integer. The integers a and b are called congruent modulo n if they have the same

More information

Relative Densities of Ramified Primes 1 in Q( pq)

Relative Densities of Ramified Primes 1 in Q( pq) International Mathematical Forum, 3, 2008, no. 8, 375-384 Relative Densities of Ramified Primes 1 in Q( pq) Michele Elia Politecnico di Torino, Italy elia@polito.it Abstract The relative densities of rational

More information

On the Exponential Diophantine Equation a 2x + a x b y + b 2y = c z

On the Exponential Diophantine Equation a 2x + a x b y + b 2y = c z International Mathematical Forum, Vol. 8, 2013, no. 20, 957-965 HIKARI Ltd, www.m-hikari.com On the Exponential Diophantine Equation a 2x + a x b y + b 2y = c z Nobuhiro Terai Division of Information System

More information

Introduction to Number Theory

Introduction to Number Theory INTRODUCTION Definition: Natural Numbers, Integers Natural numbers: N={0,1,, }. Integers: Z={0,±1,±, }. Definition: Divisor If a Z can be writeen as a=bc where b, c Z, then we say a is divisible by b or,

More information

Mathematical Foundations of Public-Key Cryptography

Mathematical Foundations of Public-Key Cryptography Mathematical Foundations of Public-Key Cryptography Adam C. Champion and Dong Xuan CSE 4471: Information Security Material based on (Stallings, 2006) and (Paar and Pelzl, 2010) Outline Review: Basic Mathematical

More information

A LOWER BOUND FOR THE SIZE OF A MINKOWSKI SUM OF DILATES. 1. Introduction

A LOWER BOUND FOR THE SIZE OF A MINKOWSKI SUM OF DILATES. 1. Introduction A LOWER BOUND FOR THE SIZE OF A MINKOWSKI SUM OF DILATES Y. O. HAMIDOUNE AND J. RUÉ Abstract. Let A be a finite nonempty set of integers. An asymptotic estimate of several dilates sum size was obtained

More information

The primitive root theorem

The primitive root theorem The primitive root theorem Mar Steinberger First recall that if R is a ring, then a R is a unit if there exists b R with ab = ba = 1. The collection of all units in R is denoted R and forms a group under

More information

Colloq. Math. 145(2016), no. 1, ON SOME UNIVERSAL SUMS OF GENERALIZED POLYGONAL NUMBERS. 1. Introduction. x(x 1) (1.1) p m (x) = (m 2) + x.

Colloq. Math. 145(2016), no. 1, ON SOME UNIVERSAL SUMS OF GENERALIZED POLYGONAL NUMBERS. 1. Introduction. x(x 1) (1.1) p m (x) = (m 2) + x. Colloq. Math. 145(016), no. 1, 149-155. ON SOME UNIVERSAL SUMS OF GENERALIZED POLYGONAL NUMBERS FAN GE AND ZHI-WEI SUN Abstract. For m = 3, 4,... those p m (x) = (m )x(x 1)/ + x with x Z are called generalized

More information

arxiv: v1 [math.gm] 1 Oct 2015

arxiv: v1 [math.gm] 1 Oct 2015 A WINDOW TO THE CONVERGENCE OF A COLLATZ SEQUENCE arxiv:1510.0174v1 [math.gm] 1 Oct 015 Maya Mohsin Ahmed maya.ahmed@gmail.com Accepted: Abstract In this article, we reduce the unsolved problem of convergence

More information

Definitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch

Definitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch Definitions, Theorems and Exercises Abstract Algebra Math 332 Ethan D. Bloch December 26, 2013 ii Contents 1 Binary Operations 3 1.1 Binary Operations............................... 4 1.2 Isomorphic Binary

More information

arxiv:math/ v1 [math.nt] 9 Aug 2004

arxiv:math/ v1 [math.nt] 9 Aug 2004 arxiv:math/0408107v1 [math.nt] 9 Aug 2004 ELEMENTARY RESULTS ON THE BINARY QUADRATIC FORM a 2 + ab + b 2 UMESH P. NAIR Abstract. This paper examines with elementary proofs some interesting properties of

More information

Discrete Math, Second Problem Set (June 24)

Discrete Math, Second Problem Set (June 24) Discrete Math, Second Problem Set (June 24) REU 2003 Instructor: Laszlo Babai Scribe: D Jeremy Copeland 1 Number Theory Remark 11 For an arithmetic progression, a 0, a 1 = a 0 +d, a 2 = a 0 +2d, to have

More information

GALOIS GROUPS OF CUBICS AND QUARTICS (NOT IN CHARACTERISTIC 2)

GALOIS GROUPS OF CUBICS AND QUARTICS (NOT IN CHARACTERISTIC 2) GALOIS GROUPS OF CUBICS AND QUARTICS (NOT IN CHARACTERISTIC 2) KEITH CONRAD We will describe a procedure for figuring out the Galois groups of separable irreducible polynomials in degrees 3 and 4 over

More information

On the Rank of the Elliptic Curve y 2 = x 3 nx

On the Rank of the Elliptic Curve y 2 = x 3 nx International Journal of Algebra, Vol. 6, 2012, no. 18, 885-901 On the Rank of the Elliptic Curve y 2 = x 3 nx Yasutsugu Fujita College of Industrial Technology, Nihon University 2-11-1 Shin-ei, Narashino,

More information

Prime Numbers and Irrational Numbers

Prime Numbers and Irrational Numbers Chapter 4 Prime Numbers and Irrational Numbers Abstract The question of the existence of prime numbers in intervals is treated using the approximation of cardinal of the primes π(x) given by Lagrange.

More information

MATRIX GENERATION OF THE DIOPHANTINE SOLUTIONS TO SUMS OF 3 n 9 SQUARES THAT ARE SQUARE (PREPRINT) Abstract

MATRIX GENERATION OF THE DIOPHANTINE SOLUTIONS TO SUMS OF 3 n 9 SQUARES THAT ARE SQUARE (PREPRINT) Abstract MATRIX GENERATION OF THE DIOPHANTINE SOLUTIONS TO SUMS OF 3 n 9 SQUARES THAT ARE SQUARE (PREPRINT) JORDAN O. TIRRELL Student, Lafayette College Easton, PA 18042 USA e-mail: tirrellj@lafayette.edu CLIFFORD

More information

A Variation of a Congruence of Subbarao for n = 2 α 5 β, α 0, β 0

A Variation of a Congruence of Subbarao for n = 2 α 5 β, α 0, β 0 Introduction A Variation of a Congruence of Subbarao for n = 2 α 5 β, α 0, β 0 Diophantine Approximation and Related Topics, 2015 Aarhus, Denmark Sanda Bujačić 1 1 Department of Mathematics University

More information

Some Highlights along a Path to Elliptic Curves

Some Highlights along a Path to Elliptic Curves 11/8/016 Some Highlights along a Path to Elliptic Curves Part : Conic Sections and Rational Points Steven J Wilson, Fall 016 Outline of the Series 1 The World of Algebraic Curves Conic Sections and Rational

More information

LEGENDRE S THEOREM, LEGRANGE S DESCENT

LEGENDRE S THEOREM, LEGRANGE S DESCENT LEGENDRE S THEOREM, LEGRANGE S DESCENT SUPPLEMENT FOR MATH 370: NUMBER THEORY Abstract. Legendre gave simple necessary and sufficient conditions for the solvablility of the diophantine equation ax 2 +

More information

Right Tetrahedra and Pythagorean Quadruples

Right Tetrahedra and Pythagorean Quadruples The Minnesota Journal of Undergraduate Mathematics Right Tetrahedra and Pythagorean Quadruples Shrijana Gurung Minnesota State University Moorhead The Minnesota Journal of Undergraduate Mathematics Volume

More information

2018 MOAA Gunga Bowl: Solutions

2018 MOAA Gunga Bowl: Solutions 08 MOAA Gunga Bowl: Solutions Andover Math Club Set Solutions. We pair up the terms as follows: +++4+5+6+7+8+9+0+ = (+)+(+0)+(+9)+(4+8)+(5+7)+6 = 5+6 = 66.. By the difference of squares formula, 6 x =

More information

2016 OHMIO Individual Competition

2016 OHMIO Individual Competition 06 OHMIO Individual Competition. Taylor thought of three positive integers a, b, c, all between and 0 inclusive. The three integers form a geometric sequence. Taylor then found the number of positive integer

More information

Diophantine Equations and Hilbert s Theorem 90

Diophantine Equations and Hilbert s Theorem 90 Diophantine Equations and Hilbert s Theorem 90 By Shin-ichi Katayama Department of Mathematical Sciences, Faculty of Integrated Arts and Sciences The University of Tokushima, Minamijosanjima-cho 1-1, Tokushima

More information

On the Partitions of a Number into Arithmetic Progressions

On the Partitions of a Number into Arithmetic Progressions 1 3 47 6 3 11 Journal of Integer Sequences, Vol 11 (008), Article 0854 On the Partitions of a Number into Arithmetic Progressions Augustine O Munagi and Temba Shonhiwa The John Knopfmacher Centre for Applicable

More information

1 2 3 style total. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points.

1 2 3 style total. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points. 1 2 3 style total Math 415 Examination 3 Please print your name: Answer Key 1 True/false Circle the correct answer; no explanation is required. Each problem in this section counts 5 points. 1. The rings

More information

A Pellian equation with primes and applications to D( 1)-quadruples

A Pellian equation with primes and applications to D( 1)-quadruples A Pellian equation with primes and applications to D( 1)-quadruples Andrej Dujella 1, Mirela Jukić Bokun 2 and Ivan Soldo 2, 1 Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička

More information

Rational Points on Conics, and Local-Global Relations in Number Theory

Rational Points on Conics, and Local-Global Relations in Number Theory Rational Points on Conics, and Local-Global Relations in Number Theory Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu http://www.math.purdue.edu/ lipman November 26, 2007

More information

School of Mathematics and Statistics. MT5836 Galois Theory. Handout 0: Course Information

School of Mathematics and Statistics. MT5836 Galois Theory. Handout 0: Course Information MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout 0: Course Information Lecturer: Martyn Quick, Room 326. Prerequisite: MT3505 (or MT4517) Rings & Fields Lectures: Tutorials: Mon

More information

Baltic Way 2008 Gdańsk, November 8, 2008

Baltic Way 2008 Gdańsk, November 8, 2008 Baltic Way 008 Gdańsk, November 8, 008 Problems and solutions Problem 1. Determine all polynomials p(x) with real coefficients such that p((x + 1) ) = (p(x) + 1) and p(0) = 0. Answer: p(x) = x. Solution:

More information

Explicit Factorizations of Cyclotomic and Dickson Polynomials over Finite Fields

Explicit Factorizations of Cyclotomic and Dickson Polynomials over Finite Fields Southern Illinois University Carbondale OpenSIUC Articles and Preprints Department of Mathematics 2007 Explicit Factorizations of Cyclotomic and Dickson Polynomials over Finite Fields Robert W. Fitzgerald

More information

On the indecomposability of polynomials

On the indecomposability of polynomials On the indecomposability of polynomials Andrej Dujella, Ivica Gusić and Robert F. Tichy Abstract Applying a combinatorial lemma a new sufficient condition for the indecomposability of integer polynomials

More information

HMMT February 2018 February 10, 2018

HMMT February 2018 February 10, 2018 HMMT February 018 February 10, 018 Algebra and Number Theory 1. For some real number c, the graphs of the equation y = x 0 + x + 18 and the line y = x + c intersect at exactly one point. What is c? 18

More information

Primes of the form ±a 2 ± qb 2

Primes of the form ±a 2 ± qb 2 Stud. Univ. Babeş-Bolyai Math. 58(2013), No. 4, 421 430 Primes of the form ±a 2 ± qb 2 Eugen J. Ionascu and Jeff Patterson To the memory of Professor Mircea-Eugen Craioveanu (1942-2012) Abstract. Reresentations

More information

Congruent Number Problem and Elliptic curves

Congruent Number Problem and Elliptic curves Congruent Number Problem and Elliptic curves December 12, 2010 Contents 1 Congruent Number problem 2 1.1 1 is not a congruent number.................................. 2 2 Certain Elliptic Curves 4 3 Using

More information

SF2729 GROUPS AND RINGS LECTURE NOTES

SF2729 GROUPS AND RINGS LECTURE NOTES SF2729 GROUPS AND RINGS LECTURE NOTES 2011-03-01 MATS BOIJ 6. THE SIXTH LECTURE - GROUP ACTIONS In the sixth lecture we study what happens when groups acts on sets. 1 Recall that we have already when looking

More information

PUTNAM TRAINING NUMBER THEORY. Exercises 1. Show that the sum of two consecutive primes is never twice a prime.

PUTNAM TRAINING NUMBER THEORY. Exercises 1. Show that the sum of two consecutive primes is never twice a prime. PUTNAM TRAINING NUMBER THEORY (Last updated: December 11, 2017) Remark. This is a list of exercises on Number Theory. Miguel A. Lerma Exercises 1. Show that the sum of two consecutive primes is never twice

More information

Cullen Numbers in Binary Recurrent Sequences

Cullen Numbers in Binary Recurrent Sequences Cullen Numbers in Binary Recurrent Sequences Florian Luca 1 and Pantelimon Stănică 2 1 IMATE-UNAM, Ap. Postal 61-3 (Xangari), CP 58 089 Morelia, Michoacán, Mexico; e-mail: fluca@matmor.unam.mx 2 Auburn

More information

EGMO 2016, Day 1 Solutions. Problem 1. Let n be an odd positive integer, and let x1, x2,..., xn be non-negative real numbers.

EGMO 2016, Day 1 Solutions. Problem 1. Let n be an odd positive integer, and let x1, x2,..., xn be non-negative real numbers. EGMO 2016, Day 1 Solutions Problem 1. Let n be an odd positive integer, and let x1, x2,..., xn be non-negative real numbers. Show that min (x2i + x2i+1 ) max (2xj xj+1 ), j=1,...,n i=1,...,n where xn+1

More information

a 2 + b 2 = (p 2 q 2 ) 2 + 4p 2 q 2 = (p 2 + q 2 ) 2 = c 2,

a 2 + b 2 = (p 2 q 2 ) 2 + 4p 2 q 2 = (p 2 + q 2 ) 2 = c 2, 5.3. Pythagorean triples Definition. A Pythagorean triple is a set (a, b, c) of three integers such that (in order) a 2 + b 2 c 2. We may as well suppose that all of a, b, c are non-zero, and positive.

More information

Pencils of Quadratic Forms over Finite Fields

Pencils of Quadratic Forms over Finite Fields Southern Illinois University Carbondale OpenSIUC Articles and Preprints Department of Mathematics 2004 Pencils of Quadratic Forms over Finite Fields Robert W. Fitzgerald Southern Illinois University Carbondale,

More information

CCSU Regional Math Competition, 2012 Part I

CCSU Regional Math Competition, 2012 Part I CCSU Regional Math Competition, Part I Each problem is worth ten points. Please be sure to use separate pages to write your solution for every problem.. Find all real numbers r, with r, such that a -by-r

More information

The Chinese Remainder Theorem

The Chinese Remainder Theorem Chapter 5 The Chinese Remainder Theorem 5.1 Coprime moduli Theorem 5.1. Suppose m, n N, and gcd(m, n) = 1. Given any remainders r mod m and s mod n we can find N such that N r mod m and N s mod n. Moreover,

More information

NOTES ON SIMPLE NUMBER THEORY

NOTES ON SIMPLE NUMBER THEORY NOTES ON SIMPLE NUMBER THEORY DAMIEN PITMAN 1. Definitions & Theorems Definition: We say d divides m iff d is positive integer and m is an integer and there is an integer q such that m = dq. In this case,

More information

SUMS PROBLEM COMPETITION, 2000

SUMS PROBLEM COMPETITION, 2000 SUMS ROBLEM COMETITION, 2000 SOLUTIONS 1 The result is well known, and called Morley s Theorem Many proofs are known See for example HSM Coxeter, Introduction to Geometry, page 23 2 If the number of vertices,

More information

2.3 In modular arithmetic, all arithmetic operations are performed modulo some integer.

2.3 In modular arithmetic, all arithmetic operations are performed modulo some integer. CHAPTER 2 INTRODUCTION TO NUMBER THEORY ANSWERS TO QUESTIONS 2.1 A nonzero b is a divisor of a if a = mb for some m, where a, b, and m are integers. That is, b is a divisor of a if there is no remainder

More information

The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Euclid Contest. Thursday, April 6, 2017

The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Euclid Contest. Thursday, April 6, 2017 The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca 017 Euclid Contest Thursday, April 6, 017 (in North America and South America) Friday, April 7, 017 (outside of North America and

More information

1 x i. i=1 EVEN NUMBERS RAFAEL ARCE-NAZARIO, FRANCIS N. CASTRO, AND RAÚL FIGUEROA

1 x i. i=1 EVEN NUMBERS RAFAEL ARCE-NAZARIO, FRANCIS N. CASTRO, AND RAÚL FIGUEROA Volume, Number 2, Pages 63 78 ISSN 75-0868 ON THE EQUATION n i= = IN DISTINCT ODD OR EVEN NUMBERS RAFAEL ARCE-NAZARIO, FRANCIS N. CASTRO, AND RAÚL FIGUEROA Abstract. In this paper we combine theoretical

More information

CYCLES AND FIXED POINTS OF HAPPY FUNCTIONS

CYCLES AND FIXED POINTS OF HAPPY FUNCTIONS Journal of Combinatorics and Number Theory Volume 2, Issue 3, pp. 65-77 ISSN 1942-5600 c 2010 Nova Science Publishers, Inc. CYCLES AND FIXED POINTS OF HAPPY FUNCTIONS Kathryn Hargreaves and Samir Siksek

More information

Decomposition of a recursive family of polynomials

Decomposition of a recursive family of polynomials Decomposition of a recursive family of polynomials Andrej Dujella and Ivica Gusić Abstract We describe decomposition of polynomials f n := f n,b,a defined by f 0 := B, f 1 (x := x, f n+1 (x = xf n (x af

More information

Perspectives of Mathematics

Perspectives of Mathematics Perspectives of Mathematics Module on Diophantine Equations and Geometry Fall 2004 Michael Stoll What is a diophantine equation? 1. Introduction and Examples Being diophantine is not so much a property

More information

A Note on Indefinite Ternary Quadratic Forms Representing All Odd Integers. Key Words: Quadratic Forms, indefinite ternary quadratic.

A Note on Indefinite Ternary Quadratic Forms Representing All Odd Integers. Key Words: Quadratic Forms, indefinite ternary quadratic. Bol. Soc. Paran. Mat. (3s.) v. 23 1-2 (2005): 85 92. c SPM ISNN-00378712 A Note on Indefinite Ternary Quadratic Forms Representing All Odd Integers Jean Bureau and Jorge Morales abstract: In this paper

More information

Pythagorean Triangles with Repeated Digits Different Bases

Pythagorean Triangles with Repeated Digits Different Bases arxiv:0908.3866v1 [math.gm] 6 Aug 009 Pythagorean Triangles with Repeated Digits Different Bases Habib Muzaffar Department of Mathematics International Islamic University Islamabad P.O. Box 144, Islamabad,

More information

A FIRST COURSE IN NUMBER THEORY

A FIRST COURSE IN NUMBER THEORY A FIRST COURSE IN NUMBER THEORY ALEXANDRU BUIUM Contents 1. Introduction 2 2. The integers 4 3. Induction 6 4. Finite sets, finite sums, finite products 7 5. The rationals 8 6. Divisibility and Euclid

More information

Math 2070BC Term 2 Weeks 1 13 Lecture Notes

Math 2070BC Term 2 Weeks 1 13 Lecture Notes Math 2070BC 2017 18 Term 2 Weeks 1 13 Lecture Notes Keywords: group operation multiplication associative identity element inverse commutative abelian group Special Linear Group order infinite order cyclic

More information

Stab(t) = {h G h t = t} = {h G h (g s) = g s} = {h G (g 1 hg) s = s} = g{k G k s = s} g 1 = g Stab(s)g 1.

Stab(t) = {h G h t = t} = {h G h (g s) = g s} = {h G (g 1 hg) s = s} = g{k G k s = s} g 1 = g Stab(s)g 1. 1. Group Theory II In this section we consider groups operating on sets. This is not particularly new. For example, the permutation group S n acts on the subset N n = {1, 2,...,n} of N. Also the group

More information

ON UNIVERSAL SUMS OF POLYGONAL NUMBERS

ON UNIVERSAL SUMS OF POLYGONAL NUMBERS Sci. China Math. 58(2015), no. 7, 1367 1396. ON UNIVERSAL SUMS OF POLYGONAL NUMBERS Zhi-Wei SUN Department of Mathematics, Nanjing University Nanjing 210093, People s Republic of China zwsun@nju.edu.cn

More information

MATH 361: NUMBER THEORY FOURTH LECTURE

MATH 361: NUMBER THEORY FOURTH LECTURE MATH 361: NUMBER THEORY FOURTH LECTURE 1. Introduction Everybody knows that three hours after 10:00, the time is 1:00. That is, everybody is familiar with modular arithmetic, the usual arithmetic of the

More information

THE TRIANGULAR THEOREM OF THE PRIMES : BINARY QUADRATIC FORMS AND PRIMITIVE PYTHAGOREAN TRIPLES

THE TRIANGULAR THEOREM OF THE PRIMES : BINARY QUADRATIC FORMS AND PRIMITIVE PYTHAGOREAN TRIPLES THE TRIANGULAR THEOREM OF THE PRIMES : BINARY QUADRATIC FORMS AND PRIMITIVE PYTHAGOREAN TRIPLES Abstract. This article reports the occurrence of binary quadratic forms in primitive Pythagorean triangles

More information

Chapter 4. Characters and Gauss sums. 4.1 Characters on finite abelian groups

Chapter 4. Characters and Gauss sums. 4.1 Characters on finite abelian groups Chapter 4 Characters and Gauss sums 4.1 Characters on finite abelian groups In what follows, abelian groups are multiplicatively written, and the unit element of an abelian group A is denoted by 1 or 1

More information

Fermat numbers and integers of the form a k + a l + p α

Fermat numbers and integers of the form a k + a l + p α ACTA ARITHMETICA * (200*) Fermat numbers and integers of the form a k + a l + p α by Yong-Gao Chen (Nanjing), Rui Feng (Nanjing) and Nicolas Templier (Montpellier) 1. Introduction. In 1849, A. de Polignac

More information

k, then n = p2α 1 1 pα k

k, then n = p2α 1 1 pα k Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

More information

An Additive Characterization of Fibers of Characters on F p

An Additive Characterization of Fibers of Characters on F p An Additive Characterization of Fibers of Characters on F p Chris Monico Texas Tech University Lubbock, TX c.monico@ttu.edu Michele Elia Politecnico di Torino Torino, Italy elia@polito.it January 30, 2009

More information

Wilson s Theorem and Fermat s Little Theorem

Wilson s Theorem and Fermat s Little Theorem Wilson s Theorem and Fermat s Little Theorem Wilson stheorem THEOREM 1 (Wilson s Theorem): (p 1)! 1 (mod p) if and only if p is prime. EXAMPLE: We have (2 1)!+1 = 2 (3 1)!+1 = 3 (4 1)!+1 = 7 (5 1)!+1 =

More information

MT5836 Galois Theory MRQ

MT5836 Galois Theory MRQ MT5836 Galois Theory MRQ May 3, 2017 Contents Introduction 3 Structure of the lecture course............................... 4 Recommended texts..................................... 4 1 Rings, Fields and

More information

Quadratic reciprocity and the Jacobi symbol Stephen McAdam Department of Mathematics University of Texas at Austin

Quadratic reciprocity and the Jacobi symbol Stephen McAdam Department of Mathematics University of Texas at Austin Quadratic reciprocity and the Jacobi symbol Stephen McAdam Department of Mathematics University of Texas at Austin mcadam@math.utexas.edu Abstract: We offer a proof of quadratic reciprocity that arises

More information

Heron triangles which cannot be decomposed into two integer right triangles

Heron triangles which cannot be decomposed into two integer right triangles Heron triangles which cannot be decomposed into two integer right triangles Paul Yiu Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida 33431 yiu@fau.edu 41st Meeting

More information

Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.

Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3

More information

TC10 / 3. Finite fields S. Xambó

TC10 / 3. Finite fields S. Xambó TC10 / 3. Finite fields S. Xambó The ring Construction of finite fields The Frobenius automorphism Splitting field of a polynomial Structure of the multiplicative group of a finite field Structure of the

More information

Know the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element.

Know the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element. The first exam will be on Monday, June 8, 202. The syllabus will be sections. and.2 in Lax, and the number theory handout found on the class web site, plus the handout on the method of successive squaring

More information

x 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?

x 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line? 1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number

More information

Course 2316 Sample Paper 1

Course 2316 Sample Paper 1 Course 2316 Sample Paper 1 Timothy Murphy April 19, 2015 Attempt 5 questions. All carry the same mark. 1. State and prove the Fundamental Theorem of Arithmetic (for N). Prove that there are an infinity

More information

Tripotents: a class of strongly clean elements in rings

Tripotents: a class of strongly clean elements in rings DOI: 0.2478/auom-208-0003 An. Şt. Univ. Ovidius Constanţa Vol. 26(),208, 69 80 Tripotents: a class of strongly clean elements in rings Grigore Călugăreanu Abstract Periodic elements in a ring generate

More information

Math 118: Advanced Number Theory. Samit Dasgupta and Gary Kirby

Math 118: Advanced Number Theory. Samit Dasgupta and Gary Kirby Math 8: Advanced Number Theory Samit Dasgupta and Gary Kirby April, 05 Contents Basics of Number Theory. The Fundamental Theorem of Arithmetic......................... The Euclidean Algorithm and Unique

More information

D( 1)-QUADRUPLES AND PRODUCTS OF TWO PRIMES. Anitha Srinivasan Saint Louis University-Madrid campus, Spain

D( 1)-QUADRUPLES AND PRODUCTS OF TWO PRIMES. Anitha Srinivasan Saint Louis University-Madrid campus, Spain GLASNIK MATEMATIČKI Vol. 50(70)(2015), 261 268 D( 1)-QUADRUPLES AND PRODUCTS OF TWO PRIMES Anitha Srinivasan Saint Louis University-Madrid campus, Spain Abstract. A D( 1)-quadruple is a set of positive

More information

Some examples of two-dimensional regular rings

Some examples of two-dimensional regular rings Bull. Math. Soc. Sci. Math. Roumanie Tome 57(105) No. 3, 2014, 271 277 Some examples of two-dimensional regular rings by 1 Tiberiu Dumitrescu and 2 Cristodor Ionescu Abstract Let B be a ring and A = B[X,

More information

2016 AMC 12A. 3 The remainder can be defined for all real numbers x and y with y 0 by x rem(x,y) = x y y

2016 AMC 12A. 3 The remainder can be defined for all real numbers x and y with y 0 by x rem(x,y) = x y y What is the value of! 0!? 9! (A) 99 (B) 00 (C) 0 (D) (E) For what value of x does 0 x 00 x = 000 5? (A) (B) (C) (D) 4 (E) 5 The remainder can be defined for all real numbers x and y with y 0 by x rem(x,y)

More information

School of Mathematics

School of Mathematics School of Mathematics Programmes in the School of Mathematics Programmes including Mathematics Final Examination Final Examination 06 22498 MSM3P05 Level H Number Theory 06 16214 MSM4P05 Level M Number

More information

. In particular if a b then N(

. In particular if a b then N( Gaussian Integers II Let us summarise what we now about Gaussian integers so far: If a, b Z[ i], then N( ab) N( a) N( b). In particular if a b then N( a ) N( b). Let z Z[i]. If N( z ) is an integer prime,

More information